Some Continuum Percolation Results

Abstract

The intent of the present chapter is to derive and discuss some basic results and specific developments in continuum percolation theory. We will begin with a discussion of exact results for cluster statistics and other percolation descriptors for a prototypical model of continuum percolation, namely, identical overlapping spheres in d dimensions. Subsequently, we will describe an Ornstein-Zernike formalism to find the pair-connectedness function P2(r) for general isotropic models of continuum percolation. The reader should note the beautiful correspondence of this theory to the Ornstein-Zernike formalism for the total correlation function h(r) of equilibrium (or thermal) systems discussed in Chapter 3. This will be followed by a discussion of various approximation schemes to close the resulting integral equation, including the Percus-Yevick approximation. The next topic will be the two-point cluster function C2(r). First we will present an exact series representation of C2(r) for dispersions and then discuss its analytical evaluation for certain models. The chapter will conclude with a presentation of percolation thresholds for overlapping sphere systems, overlapping particles of nonspherical shape, and interacting particle systems. The reader is referred to Meester and Roy (1996) for a more mathematical treatment of continuum percolation.

Keywords

Percolation Threshold Interact Particle System Average Coordination Number Continuum Percolation Total Correlation Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.